Question: You have found the following ages (in years) of all 4 seals at your local zoo: $ 8,\enspace 7,\enspace 3,\enspace 11$ What is the average age of the seals at your zoo? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we have data for all 4 seals at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{8 + 7 + 3 + 11}{{4}} = {7.3\text{ years old}} $ Find the squared deviations from the mean for each seal. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $8$ years $0.7$ years $0.49$ years $^2$ $7$ years $-0.3$ years $0.09$ years $^2$ $3$ years $-4.3$ years $18.49$ years $^2$ $11$ years $3.7$ years $13.69$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{0.49} + {0.09} + {18.49} + {13.69}} {{4}} $ $ {\sigma^2} = \dfrac{{32.76}}{{4}} = {8.19\text{ years}^2} $ The average seal at the zoo is 7.3 years old. The population variance is 8.19 years $^2$.